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10. Sprouts

This is more of an activity than a puzzle and is really a game for two players.

Start with three dots.

Each player takes it turn to join any two dots and adding an extra dot on the line that joins them. The first player, then, might join dots like this:

The second player might respond with:

At this stage we should note that lines are not allowed to cross and no point can have more more than three lines going to it. The game continues until no more points can be joined and the last player to go wins.

A possible continuation of the game is as follows:

So player 2 wins. Is this always the case? What about starting with four dots? What if you allow points to have four lines going to them instead of three?


 

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